Counting and Sequential Information Processing in Mechanical Metamaterials

Materials with an irreversible response to cyclic driving exhibit an evolving internal state which, in principle, encodes information on the driving history. Here we realize irreversible metamaterials that count mechanical driving cycles and store the result into easily interpretable internal states. We extend these designs to aperiodic metamaterials that are sensitive to the order of different driving magnitudes, and realize “lock and key” metamaterials that only reach a specific state for a given target driving sequence. Our metamaterials are robust, scalable, and extendable, give insight into the transient memories of complex media, and open new routes towards smart sensing, soft robotics, and mechanical information processing.

Figure 1

(a) Schematic representation of the evolution of a “beam counter” metamaterial with $n=4$ unit cells under two compression cycles. Each unit cell contains a buckled $m$-beam (memory beam) which encodes a single bit. When the strain $ϵ≔\mathrm{\Delta }/L$ is cycled between ${ϵ}_m$ and ${ϵ}_M$, the $m$-beams interact (gray symbols), so that the “1” state is copied to the right (triangle), leading to the step wise advancing of the 1 state to the right. (b) Geometry of a unit cell $i$ (highlighted), containing a slender $m$-beam of width $t_i$ and a thicker, asymmetric $a$-beam (ancillary slitted beam) of width $T_i$. The lengths are nondimensionalized by setting the beams rest lengths to 1 (corners and slits highlighted by white markers). (c) Evolution of beam pairs under increased compression from ${ϵ}_m$ (left) to ${ϵ}_M$ (right): when the spacing $d_i$ is smaller than a critical distance $D^{*}$, the buckled state of the slender beam is copied to the thick beam (top); when $D_i>D^{*}$, the buckled state of the thick beam is copied to the slender beam (bottom).

Figure 2

(a) Beam counting metamaterial with 11 units, with center region highlighted; see movie 1 in [30] ($t_i=0.040$, $T_i=0.10$, $d_i=0.13$, and $D_i=0.15$). Arrows indicate weak symmetry breaking of the $m$-beams that makes the system reach state $\{10000000000\}$ when $ϵ$ is increased from zero. (b) Space-time plot, tracing the center positions of each $m$-beam as a function of time under cyclic compression ${ϵ}_m\nearrow {ϵ}_M\searrow {ϵ}_m$ (${ϵ}_m=0.026$, ${ϵ}_M=0.099$). Beams in state 0 (1) are highlighted in yellow (blue). (c) Evolution of the beam counter prepared in the initial state $\{01000111010\}$ (central parts shown only; beam state colored as above).

Figure 3

Comparison of the evolution of two unit cells during a compression cycle. (a) Original design. (b) Design without slits, which does not copy the 1 state. Frames (aIV) and (bIV) are not at the same strain, but compare the states where the second $m$-beam just loses contact with one of its neighboring $a$-beams; note that the $m$-beam is then, respectively, to the right (a) and left (b) of the neutral line (dashed).

Figure 4

Response of aggregate metamaterials containing multiple homogeneous and heterogeneous beam counters. (a) Complex driving cycles can be encoded as an input string (here $BAC$) and produce different sequential response in counters $aaa$, $bbb$, and $ccc$ (boxes represent individual counters and numbers represent the sequence of states ${\sigma }_i$). (b) Experimental realization of the $aaa|bbb|ccc$ aggregate metamaterial, and snapshots of its pathway in response to input driving $BAC$ (triangles highlight the right copying of a 1-bit). Note that $t_i$, $d_i$, and $D_i$ subtly vary along the counters while $T_i$ remains constant (see Supplemental Material [30]). The color and triangle overlay help to indicate the state and the transitions of the beams. (c) Transition graph of the heterogeneous counter $bac$. (d) Experimental realization and evolution of $bac$ counter during input string $BAC$ [panel (b) and (d) are snapshots of the same experiment]. (e) The output of the $bac$ counter to all input sequences which are permutations of $ABC$ (rounded boxes). (f) The four counters $aaa$, $bbb$, $ccc$, and $bac$ operated in parallel reach the unique state $\{0120\}$ only for input sequence $BAC$. The homogeneous counters store the count of each signal, while the heterogeneous counter is sensitive to the specific permutation.